Positive Element
In mathematics, especially functional analysis, a self-adjoint (or hermitian) element of a C*-algebra is called positive if its spectrum consists of non-negative real numbers. Moreover, an element of a C*-algebra is positive if and only if there is some in such that . A positive element is self-adjoint and thus normal.
If is a bounded linear operator on a Hilbert space, then this notion coincides with the condition that is non-negative for every vector in . Note that is real for every in if and only if ' is self-adjoint. Hence, a positive operator on a Hilbert space is always self-adjoint (and a self-adjoint everywhere defined operator on a Hilbert space is always bounded because of the Hellinger-Toeplitz theorem).
The set of positive elements of a C*-algebra forms a convex cone.
Read more about Positive Element: Positive and Positive Definite Operators, Examples, Partial Ordering Using Positivity
Famous quotes containing the words positive and/or element:
“People who talk about revolution and class struggle without referring explicitly to everyday life, without understanding what is subversive about love and what is positive in the refusal of constraints, such people have a corpse in their mouth.”
—Raoul Vaneigem (b. 1934)
“All forms of beauty, like all possible phenomena, contain an element of the eternal and an element of the transitory—of the absolute and of the particular. Absolute and eternal beauty does not exist, or rather it is only an abstraction creamed from the general surface of different beauties. The particular element in each manifestation comes from the emotions: and just as we have our own particular emotions, so we have our own beauty.”
—Charles Baudelaire (1821–1867)